1,220 research outputs found
Graph Coloring Problems and Group Connectivity
1. Group connectivity. Let A be an abelian group and let iA(G) be the smallest positive integer m such that Lm(G) is A-connected. A path P of G is a normal divalent path if all internal vertices of P are of degree 2 in G and if |E(P)|= 2, then P is not in a 3-cycle of G. Let l(G) = max{lcub}m : G has a normal divalent path of length m{rcub}. We obtain the following result. (i) If |A| β₯ 4, then iA( G) β€ l(G). (ii) If | A| β₯ 4, then iA(G) β€ |V(G)| -- Delta(G). (iii) Suppose that |A| β₯ 4 and d = diam( G). If d β€ |A| -- 1, then iA(G) β€ d; and if d β₯ |A|, then iA(G) β€ 2d -- |A| + 1. (iv) iZ 3 (G) β€ l(G) + 2. All those bounds are best possible.;2. Modulo orientation. A mod (2p + 1)-orientation D is an orientation of G such that d +D(v) = d--D(v) (mod 2p + 1) for any vertex v β V ( G). We prove that for any integer t β₯ 2, there exists a finite family F = F(p, t) of graphs that do not have a mod (2p + 1)-orientation, such that every graph G with independence number at most t either admits a mod (2p+1)-orientation or is contractible to a member in F. In particular, the graph family F(p, 2) is determined, and our results imply that every 8-edge-connected graph G with independence number at most two admits a mod 5-orientation.;3. Neighbor sum distinguishing total coloring. A proper total k-coloring &phis; of a graph G is a mapping from V(G) βͺ E(G) to {lcub}1,2, . . .,k{rcub} such that no adjacent or incident elements in V(G) βͺ E( G) receive the same color. Let m&phis;( v) denote the sum of the colors on the edges incident with the vertex v and the color on v. A proper total k-coloring of G is called neighbor sum distinguishing if m &phis;(u) β m&phis;( v) for each edge uv β E( G ). Let chitSigma(G) be the neighbor sum distinguishing total chromatic number of a graph G. Pilsniak and Wozniak conjectured that for any graph G, chitSigma( G) β€ Delta(G) + 3. We show that if G is a graph with treewidth ℓ β₯ 3 and Delta(G) β₯ 2ℓ + 3, then chitSigma( G) + ℓ -- 1. This upper bound confirms the conjecture for graphs with treewidth 3 and 4. Furthermore, when ℓ = 3 and Delta β₯ 9, we show that Delta(G)+1 β€ chit Sigma(G) β€ Delta(G)+2 and characterize graphs with equalities.;4. Star edge coloring. A star edge coloring of a graph is a proper edge coloring such that every connected 2-colored subgraph is a path with at most 3 edges. Let ch\u27st(G) be the list star chromatic index of G: the minimum s such that for every s-list assignment L for the edges, G has a star edge coloring from L. By introducing a stronger coloring, we show with a very concise proof that the upper bound of the star chromatic index of trees also holds for list star chromatic index of trees, i.e. ch\u27st( T) β€ [3Delta/2] for any tree T with maximum degree Delta. And then by applying some orientation technique we present two upper bounds for list star chromatic index of k-degenerate graphs
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